Degree Of Polynomial Approximation Research Articles

Efficiency high-order discontinuous Galerkin method for low speeds flows with time-derivative preconditioning

A local preconditioning high-order discontinuous Galerkin (DG) method is introduced to enhance the efficiency and accuracy of the density-based approach for low-speed flows. The method is based on the time-derivative preconditioning technique typically employed in finite volume methods. Traditional approximation methods that only precondition low-order derivatives lead to incompatibility of the equations. To extend the preconditioning technique to the DG method, we employ an analytic preconditioning mass matrix derived from the DG formulation of the preconditioned Navier–Stokes equations. Modified numerical flux functions and a self-adaptive local velocity truncation parameter are employed for DG systems with preconditioning. We demonstrate the efficiency and accuracy of the preconditioned DG method using explicit and implicit schemes for steady flows and a dual-time scheme for unsteady flows. The method has been implemented and used to compute a variety of flow problems, including inviscid and laminar flows, utilizing various degrees of polynomial approximations. Computations with and without preconditioning are performed to analyze the influence of spatial discretization on the accuracy and convergence of the DG solutions at low Mach numbers. Numerical results underscore the enhanced accuracy and convergence properties of the proposed method for low-speed flows, indicating significant performance improvements over traditional methods.

Approximation of intra-particle reaction/diffusion effects of immobilized enzyme system following reverse Michaelis–Menten (rMM) mechanism: third degree polynomial and Akbari–Ganji methods

Two approximate analytical expressions based on third degree polynomial and Akbari–Ganji’s method (AGM) were derived for the reaction/diffusion controlled kinetics of an immobilized enzyme (IE) systems. The approximation methods predict substrate concentration profile and effectiveness factor (eta) in a porous spherical particle. The reaction is assumed to follow reverse Michaelis–Menten (rMM) kinetics. The approximate methods predictions were comparable to that of numerical solution (using the Matlab finite difference function, bvp4c) at wide range of {phi }^{2} and {y}_{o} especially at low {phi }^{2} and high {y}_{o} (polynomial equation) and low {phi }^{2} and low {y}_{o} (AGM equation). Although the approximate solution was derived for rMM kinetics, the results can be used to describe other important enzymatic reaction kinetics such as simple Michaelis–Menten (MM) kinetics and MM with competitive product inhibition kinetics. A necessary design equation should be satisfied when using polynomial or AGM approximation for different enzyme kinetic equations. In this work, two examples of enzymatic reactions of industrial interest were studied, namely glucose-fructose isomerization follows rMM kinetics and hydrolysis of lactose follows Michaelis–Menten (MM) equation with competitive product (galactose) inhibition. Predictions of the developed third degree polynomial and AGM approximation equations agree with that of numerical solution, the percentage relative error for the effectiveness factor was less than 11 in comparison with the numerical solution. Good agreement between approximate and numerical estimations demonstrates the validity of these approximation methods.

Dispersion reduction in Feng and Wu’s IPDG method

We give an analytical justification of the determination of an important parameter involved in Feng and Wu’s solution of the Helmholtz equation by an Interior Penalty Discontinuous Galerkin (IPDG) method (Feng and Wu, 2009). This parameter was determined in this reference from a large number of numerical tests for a mesh composed of equal equilateral triangles at a fixed frequency. It plays an essential role in the reduction the dispersion error for a polynomial approximation of degree 1. We show how this justification makes it possible to slightly improve it and to handle other structured meshes. The analytical results are evaluated by several numerical experiments, showing in particular that the parameter determined in principle for equal equilateral triangles also eliminates dispersion for unstructured, well-smoothed meshes.

A 〖(2n+9)〗^thDegree Polynomial Approximation Method for Solving Eight Order Linear Boundary Value Problems

Galerking method, Collocation method and least squired method have been used for solving boundary value problems (BVPs) and from practical point of view almost any success of above mention method depends on the selection of basis functions. Also, Adomain decomposition and variational iteration methods have been applied for the solution of BVPs and required respectively calculating the Adomian’s polynomials and Lagrange multipliers. In this paper a simple method of Degree polynomial approximation solution were n is number of successive derivatives of the governing equation is suggested as an alternative of the above mentioned methods. The method consists of first obtaining from the governing equation its n successive derivatives and boundary conditions a linear differential system of equations evaluated at the boundary points. Next, an approximate solution in the form of polynomial of degree with unknown coefficients is assumed. To determine the unknown coefficients, the assumed solution is incorporate into the linear differential system which turns to be a linear system of equation with unknowns which is salvable uniquely. The method is tested for n=5 to four examples. It clear that from tables 3.1 to 3.4 and figures 3.1 to 3.4 the numerical outcomes agree with the exact solution and also better than some existing results in the literatures. it should be also noted that the accuracy of the method can be increased by increased n. All Numerical computation were performed using maple 2016 software.

An Anisotropic hp-mesh Adaptation Method for Time-Dependent Problems Based on Interpolation Error Control

We propose an efficient mesh adaptive method for the numerical solution of time-dependent partial differential equations considered in the fixed space-time cylinder varOmega times (0,T). We employ the space-time discontinuous Galerkin method which enables us to use different meshes at different time levels in a natural way. The mesh adaptive algorithm is based on control of the interpolation error in the L^infty (0,T; L^q(varOmega ))-norm. The goal is to construct a sequence of conforming triangular meshes in such a way that the interpolation error bound is under a given tolerance and the number of degrees of freedom is minimal. The resulting grids consist of anisotropic mesh elements with varying polynomial approximation degrees with respect to space. We present a theoretical framework of this approach as well as several numerical examples demonstrating the accuracy, efficiency, and applicability of the method.

A DG method for a stress formulation of the elasticity eigenproblem with strongly imposed symmetry

We introduce a pure–stress formulation of the elasticity eigenvalue problem with mixed boundary conditions. We propose an H(div)-based discontinuous Galerkin method that imposes strongly the symmetry of the stress for the discretization of the eigenproblem. Under appropriate assumptions on the mesh and the degree of polynomial approximation, we prove the spectral correctness of the discrete scheme and derive optimal rates of convergence for eigenvalues and eigenfunctions. Finally, we provide numerical examples in two and three dimensions.

Semi-analytical implicit direct time integration scheme on example of 1-D wave propagation problem

The most common approach in dynamic analysis of engineering structures and physical phenomenas consists in finite element discretization and mathematical formulation with subsequent application of direct time integration schemes. The space interpolation functions are usually the same as in static analysis. Here on example of 1-D wave propagation problem the original implicit scheme is proposed, which contains the time interval value explicitly in space interpolation function as results of analytical solution of differential equation for considered moment of time. The displacements (solution) at two previous moments of time are approximated as polynomial functions of position and accounted for as particular solutions of the differential equation. The scheme demonstrates the perfect predictable properties as to dispersion and dissipation. The crucial scheme parameter is the time interval – the lesser the interval the more correct results are obtained. Two other parameters of the scheme – space interval and the degree of polynomial approximation have minimal impact on the general behavior of solution and have influence on small zone near the front of the wave.

Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D

This paper presents a weak Galerkin finite element method, exploring polynomial approximations of various degree, for solving singularly perturbed convection-diffusion equation in 2D. On each mesh element, this method makes use of polynomials of degree k≥1 in the interior, polynomials of degree j≥1 on the boundary, vector-valued polynomials of degree l≥k−1 for the discrete weak gradient and polynomials of degree m≥k for the discrete weak convection divergence. Shishkin mesh is used in order that the method is uniformly convergent independent of the singular perturbation parameter. A special interpolation is delicately designed according to the structures of the designed method and Shishkin mesh. Then uniform convergence of optimal order is proved, as is also confirmed by the numerical experiments.

High-order approximation rates for shallow neural networks with cosine and ReLU activation functions

We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the smoothness in the spectral Barron space of the underlying function f to be approximated. We show that as the smoothness index s of f increases, shallow neural networks with ReLU k activation function obtain an improved approximation rate up to a best possible rate of O ( n − ( k + 1 ) log ⁡ ( n ) ) in L 2 , independent of the dimension d . The significance of this result is that the activation function ReLU k is fixed independent of the dimension, while for classical methods the degree of polynomial approximation or the smoothness of the wavelets used would have to increase in order to take advantage of the dimension dependent smoothness of f . In addition, we derive improved approximation rates for shallow neural networks with cosine activation function on the spectral Barron space. Finally, we prove lower bounds showing that the approximation rates attained are optimal under the given assumptions.

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ⟩=∑j=1rcj|φj⟩ for cj∈C and stabilizer states φj. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the n-th tensor power of single-qubit magic states.We prove a lower bound of Ω(n) on the stabilizer rank of such states, improving a previous lower bound of Ω(n) of Bravyi, Smith and Smolin \cite{BSS16}. Further, we prove that for a sufficiently small constant δ, the stabilizer rank of any state which is δ-close to those states is Ω(n/log⁡n). This is the first non-trivial lower bound for approximate stabilizer rank.Our techniques rely on the representation of stabilizer states as quadratic functions over affine subspaces of F2n, and we use tools from analysis of boolean functions and complexity theory. The proof of the first result involves a careful analysis of directional derivatives of quadratic polynomials, whereas the proof of the second result uses Razborov-Smolensky low degree polynomial approximations and correlation bounds against the majority function.

A p-robust polygonal discontinuous Galerkin method with minus one stabilization

In this paper, we introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree [Formula: see text]. The stabilization is obtained by penalizing, in each mesh element [Formula: see text], a residual in the norm of the dual of [Formula: see text]. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a [Formula: see text]-explicit stability and error analysis, proving [Formula: see text]-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement

In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with $hp$-refinement. These preconditioners are based on the decomposition of the DG finite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refined technique, which in this work we extend to the $hp$-refinement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh $h$, and the degree of polynomial approximation $p$. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refined meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2).

Sub-Optimal Convergence of Discontinuous Galerkin Methods with Central Fluxes for Linear Hyperbolic Equations with Even Degree Polynomial Approximations

Sub-Optimal Convergence of Discontinuous Galerkin Methods with Central Fluxes for Linear Hyperbolic Equations with Even Degree Polynomial Approximations

How much faster does the best polynomial approximation converge than Legendre projection?

We compare the convergence behavior of best polynomial approximations and Legendre and Chebyshev projections and derive optimal rates of convergence of Legendre projections for analytic and differentiable functions in the maximum norm. For analytic functions, we show that the best polynomial approximation of degree n is better than the Legendre projection of the same degree by a factor of $$n^{1/2}$$ . For differentiable functions such as piecewise analytic functions and functions of fractional smoothness, however, we show that the best approximation is better than the Legendre projection by only some constant factors. Our results provide some new insights into the approximation power of Legendre projections.

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

This paper presents HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator Gmsh is provided to facilitate its application to practical engineering problems.

An efficient methodology for stress‐based finite element approximations in two‐dimensional elasticity

SummaryWe present an efficient approach to compute the matrices used in finite element formulations where self‐equilibrated (zero divergence) approximations of the stress field are used. The fundamental aspect of this approach is that it is applicable to polynomial approximations of high degree (it was tested up to degree 10), providing closed‐form expressions for the elementary matrices involved. The components of the stress field are defined as a function of the coordinates in the master element, facilitating the postprocessing of the finite element results and allowing for the consideration of geometric parameters in reduced order models.

A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations

In this paper, we propose a fast algorithm for the variable-order (VO) Caputo fractional derivative based on a shifted binary block partition and uniform polynomial approximations of degree r. Compared with the general direct method, the proposed algorithm can reduce the memory requirement from O(n) to O(rlogn) storage and the complexity from O(n2) to O(rnlogn) operations, where n is the number of time steps. As an application, we develop a fast finite difference method for solving a class of VO time-fractional diffusion equations. The computational workload is of O(rmnlogn) and the active memory requirement is of O(rmlogn), where m denotes the size of spatial grids. Theoretically, the unconditional stability and error analysis for the proposed fast finite difference method are given. Numerical results of one and two dimensional problems are presented to demonstrate the well performance of the proposed method.

A space-time discontinuous Galerkin method for the elastic wave equation

In this work we present a new high order space-time discretization method based on a discontinuous Galerkin paradigm for the second order visco-elastodynamics equation. After introducing the method, we show that the resulting space-time discontinuous Galerkin formulation is well-posed, stable and retains optimal rate of convergence with respect to the discretization parameters, namely the mesh size and the polynomial approximation degree. A set of two and three-dimensional numerical experiments confirms the theoretical bounds.

Interpolatory super-convergent discontinuous Galerkin methods for nonlinear reaction diffusion equations on three dimensional domains

An error analysis of a super-convergent discontinuous Galerkin method formulated in mixed form and applied to a general class of semi-linear equations is presented. To reduce the computational cost at each time step, the nonlinear term is approximated with a Lagrange interpolatory operator. Optimal convergence of order O(hk+1), for both, the primary and auxiliary variables, is obtained for polynomial approximations of degree k. Using a well known element-by-element post-processing procedure, super-convergence of the primary variable is proven. Computational aspects for an efficient implementation are discussed in detail. To avoid solving a large global nonlinear problem at each time step the symmetric Strang operator splitting method is considered as time marching scheme. To validate our error estimates, a series of numerical experiments are carried out using unstructured meshes of three dimensional domains.

Algebraic Multigrid Schemes for High-Order Nodal Discontinuous Galerkin Methods

We present algebraic multigrid (AMG) methods for the efficient solution of the linear system of equations stemming from high-order discontinuous Galerkin (DG) discretizations of second-order elliptic problems. For DG methods, standard multigrid approaches cannot be employed because of redundancy of the degrees of freedom associated to the same grid point. We present new aggregation procedures and test them in extensive two-dimensional numerical experiments to demonstrate that the proposed AMG method is uniformly convergent with respect to all of the discretization parameters, namely the mesh-size and the polynomial approximation degree.